Just Hammond

This article features just intervals created by the mechanical tonegenerator of the classical Hammond B-3 Organ model.

Design of the Hammond B-3’s Tonegenerator

Since 1935 the Hammond Organ Company’s goal was to market electromechanical organs[1] with 12-tone equally tempered (12edo) tuning. The mechanical tonegenerator of the Hammond B-3 Organ is based on a set of twelve different pairings of gearwheels that make (12*4) driven shafts turn. The corresponding driving gearwheels are mounted on a common shaft and turn all at the same rotational speed n1. Certain gears reduce, others increase rotational speed.[2]

For every chromatic pitch class four driven shafts are installed. Pure octaves are generated by dedicated tonewheels (with 2, 4, 8, 16, 32, 64 or 128 high and low points on their edges) that rotate with the driven shafts. Each high point on a tone wheel is called a tooth. When the gears are in motion, magnetic pickups react to the tonewheels’ passing teeth and generate an electric signal that can be amplified and transmitted to a loudspeaker.

For each pair of gearwheels the ratio of rotational speed n2/n1 is determined by the inverse ratio of the gearwheels’ integer teeth numbers Z1 and Z2:

$\frac{Z_1}{Z_2}=\frac{n_2}{n_1}$

To calculate the rotational speed n2 of the driven shafts we write

$n_2=\frac{Z_1}{Z_2}\cdot n_1$

Table 1: Pairings of Gearwheels[3] / Ratios and Intervals

Pitch

Class

HAMMOND
Organ
Mechanical
Gearing

Gear Ratio
(canceled)

Conversion
of raw
ratio to
interval

Deviation
from
12tone equal
temperament
(12edo)

Intonation
(Note A renders
standard pitch,
if shaft (A) is
rotating
@20 rev./sec)

(A) (B) (C) (D)
driving

Z1[teeth]

driven

Z2[teeth]

Fraction
Ratio

(C)/(D)

[cents]

[cents]

[cents]
C 85 104 85 104 0.8173077 -349.26 -49.26 -0.576
C# 71 82 71 82 0.8658537 -249.37 -49.37 -0.684
D 67 73 67 73 0.9178082 -148.48 -48.48 0.200
D# 105 108 35 36 0.9722222 -48.77 -48.77 -0.088
E 103 100 103 100 1.0300000 51.17 -48.83 -0.145
F 84 77 12 11 1.0909091 150.64 -49.36 -0.681
F# 74 64 37 32 1.1562500 251.34 -48.66 0.026
G 98 80 49 40 1.2250000 351.34 -48.66 0.020
G# 96 74 48 37 1.2972973 450.61 -49.39 -0.707
A 88 64 11 8 1.3750000 551.32 -48.68 0.000
A# 67 46 67 46 1.4565217 651.03 -48.97 -0.285
B 108 70 54 35 1.5428571 750.73 -49.27 -0.593

(Purple colored cells contain prime numbers)

Just Intervals

When we associate “ratios of the gearwheels’ integer teeth numbers” with “frequency ratios between partials” we realize an intrinsic just interval determined by integer teeth numbers within such mechanical gear - even without turning the shafts! Although the Hammond Organ pretends to generate a 12edo scale, the instrument in fact creates a high prime limit just scale.

Tuning

The whole set of frequency ratios is fixed by the design of the gear mechanism. The driving shaft’s (A) rotational speed n1 determines the instrument’s (master-) tuning. Rotating at exactly 1200 rpm (20 rev./sec), the pitch of note A equals precisely 27.500 Hz or one of its doublings. Therefore the instrument aligns note A with a concert pitch of 440.0 Hz.

$f_\text{A}=20.0/\text{s}\cdot\frac{88}{64}\cdot(2^4)=440.0/\text{s} = 440.0 \text{ Hz}$

Mapping Hammond’s Rational Intervals to the Harmonic Series

To find out, where the rational intervals played on a Hammond Organ occur in the harmonic series we

• cancel the fractions of gear-ratios specified by Hammond and
• calculate the least common multiple (LCM) of the denominators of "intervals of interest" by prime factorization
• With this specific LCM we recalculate the numerators of the intervals. The resulting numerators correspond to the partial numbers we are looking for.

Before we proceed, we have to agree on a numbering scheme for octaves in the harmonic series.

Numbering Octaves

We apply the scheme from the article First Five Octaves of the Harmonic Series and number the octaves as follows:

• Integer octave numbering starts with #1 for the range between the 1st and < 2nd partial
• The 2nd octave starts at partial #2 (= 21) and covers partials 2 and 3
• The 3rd octave starts at partial #4 (= 22) and covers partials 4, 5, 6 and 7
• The 4th octave starts at partial #8 (= 23) and covers partials 8, 9, 10, 11, 12, 13, 14 and 15.
• ...

This numbering scheme is consistent with the scheme used by Bill Sethares[4] : “In general, the nth octave contains 2n-1 pitches.”

Mapping Hammond’s Rational Intervals (cont.): Examples

The following examples illustrate how to map intervals or chords to the harmonic series.

Example 1: Mapping a single interval

In this example we map the combination of a Hammond Organ’s note E and a higher note A to the harmonic series.

Table 2: Mapping the fourth E-A

Pitch
Class

HAMMOND
Gear Ratio
(canceled)

Prime
Factorization

Ascending
Partial Numbers of
an Overtone Scale

Partial Found
in Octave

(C) (D)

...of Column (D)

Recalculated
Numerator =
Partial#
P=LCM *(C)/(D)

Counted
from 1/1:
Octave# =
1+(ln(P)/ln(2))

Fraction

(C)/(D)

E 103 100 2 2 5 5 206 8.7
A 11 8 2 2 2 275 9.6

Multiply -------->
to find (D)'s least
common multiple
(LCM)

2

2

2

5

5

200

(LCM)

8.6

Decimal printed
for orientation only

The resulting interval appears between partial # 206 and partial # 275. Thus the frequency ratio is (275:206), which equals 500.14 cents.

Example 2: Mapping a chord

Adding an upper fifth (note B), the second example illustrates how to map the resulting sus4-chord E-A-B to the harmonic series.

Table 3: sus4-chord E-A-B

Pitch
Class

HAMMOND
Gear Ratio
(canceled)

Prime
Factorization

Ascending
Partial Numbers of
an Overtone Scale

Partial found
in Octave

(C) (D)

...of Column (D)

Recalculated
Numerator =
Partial#
P=LCM *(C)/(D)

Counted
from 1/1:
Octave# =
1+(ln(P)/ln(2))

Fraction

(C)/(D)

E 103 100 2 2 5 5 1442 11.5
A 11 8 2 2 2 1925 11.9
B 54 35 5 7 2160 12.1

Multiply -------->
to find (D)'s least
common multiple
(LCM)

2

2

2

5

5

7

1400

(LCM)

11.4

Decimal printed
for orientation only

The supplemental note B establishes an additional prime factor. We find the matching pattern of partials for this sus4-chord (1442:1925:2160) farther up in the harmonic series, where this chord spans the boundary between the 11th and the 12th octave.

Example 3: Mapping all of the tonegenerator's pitchclasses

The full set of the Hammond Organ’s intervals resides surprisingly far up in the Harmonic Series:

• The 44th octave starts at partial #(243), just below the set of partials determined by the Hammond Organ’s tonegenerator
• The 45th octave starts right within the derived set of partials and starts at partial #(244)

Table 4: The full set of intervals' position in the Harmonic Series

Pitch
Class

HAMMOND
Gear Ratio
(canceled)

Prime
Factorization

Ascending
Partial Numbers of
an Overtone Scale

Partial Found
in Octave #

(C) (D)

... of Column (D)

Recalculated
Numerator =
Partial #
P=LCM *(C)/(D)

Counted from
1/1:
Octave# =
1+(ln(P)/ln(2))

Fraction

(C)/(D)

C 85 104 2 2 2 13 15,003,356,791,500 44.8
C# 71 82 2 41 15,894,517,438,800 44.9
D 67 73 73 16,848,249,818,400 44.9
D# 35 36 2 2 3 3 17,847,130,301,000 45.0
E 103 100 2 2 5 5 18,907,759,758,888 45.1
F 12 11 11 20,025,870,883,200 45.2
F# 37 32 2 2 2 2 2 21,225,337,107,975 45.3
G 49 40 2 2 2 5 22,487,384,179,260 45.4
G# 48 37 37 23,814,549,158,400 45.4
A 11 8 2 2 2 25,240,941,425,700 45.5
A# 67 46 2 23 26,737,439,929,200 45.6
B 54 35 5 7 28,322,303,106,240 45.7

Multiply -------->
to find (D)'s least
common multiple
(LCM)

2

2

2

2

2

3

3

5

5

7

11

13

23

37

41

73

18,357,048,309,600

(LCM)

45.1
Decimal printed
for orientation
only

Discussion

No doubt - the evidence that a cluster of 12 simultaneously ringing semitones from a Hammond Organ is allocated around the 45th octave of the harmonic series is of limited practical value. The intervals' far-up placement is mainly caused by Laurens Hammond’s implementation of various prime numbers (11, 13, 23, 37, 41, 73) in different gearwheel pairings.

• Respective high-order partials are very densely spaced (in the range of pico-cents) and intervals between successive partials up there are too narrow for musical applications by far
• Due to its construction the tonegenerator selects only twelve from 17.6 trillion varieties in the 45th octave where…
• the partial number associated with the LCM, which is located exactly 8/11 below pitch class A, is not addressed because there is no gear with transmission ratio 1.000
• no pure octave above a virtual root (1/1; partial# (244)) is playable, which would ring -624.997 cents way down from pitchclass A

General Applicability

The method of prime factorization to find the LCM can be applied to arbitrary intervals, chords or scales built from rational intervals to identify their position in the harmonic series. Simply replace the gear-ratios by just intervals of interest.

References

1. Webressource https://en.wikipedia.org/wiki/Hammond_organ (retrieved December 2019)
2. Detailed photos of a similar M-1 tonegenerator are provided by https://modularsynthesis.com/hammond/m3/m3.htm (retrieved December 2019)
3. Gearing details were taken from http://www.goodeveca.net/RotorOrgan/ToneWheelSpec.html (retrieved Dec 29, 2019) The German Wikipedia provides the same technical information (in German): https://de.wikipedia.org/wiki/Hammondorgel#Tonerzeugung (retrieved Dec 29, 2019) The HammondWiki publishes a second, alternative set of gear ratios with slightly deviating pitch class “E”. Certain other pitch classes are shifted by pure octaves. http://www.dairiki.org/HammondWiki/GearRatio (retrieved Dec 29, 2019)
4. Sethares, William A. Tuning Timbre Spectrum Scale. London: Springer Verlag , 1999.